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arxiv: 2604.27026 · v1 · submitted 2026-04-29 · 🧮 math-ph · math.AP· math.DG· math.FA· math.MP

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On the Quantisation of Linear Gauge Theories on Lorentzian Manifolds: Maxwell's Theory via Complete Gauge Fixing

Gabriel Schmid
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Pith reviewed 2026-05-07 11:20 UTC · model grok-4.3

classification 🧮 math-ph math.APmath.DGmath.FAmath.MP
keywords quantizationMaxwell theoryHadamard statesgauge fixingglobally hyperbolic spacetimesHodge decompositionSobolev spacespseudodifferential operators
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The pith

A new initial-data gauge fixing via Hodge decomposition on Cauchy surfaces allows construction of Hadamard states for the quantization of Maxwell theory on globally hyperbolic spacetimes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a quantization procedure for linear gauge theories on globally hyperbolic Lorentzian manifolds by reducing their non-hyperbolic equations to constrained hyperbolic dynamics through gauge fixing. For Maxwell's theory specifically, it applies a complete gauge fixing directly to the initial data on Cauchy surfaces, using a Hodge decomposition of differential forms in Sobolev spaces to remove unphysical degrees of freedom. States are then built with pseudodifferential calculus and shown to satisfy the Hadamard condition. A sympathetic reader cares because this yields a rigorous algebraic quantum field theory for the electromagnetic field in curved spacetime without leftover gauge artifacts or unphysical modes.

Core claim

The central claim is that a gauge-fixing procedure performed at the level of initial data, realized through a new Hodge decomposition for differential k-forms in Sobolev spaces on complete Riemannian manifolds, suppresses the unphysical degrees of freedom in Maxwell's theory. This reduction permits the construction of Hadamard states for the quantized theory on globally hyperbolic spacetimes using tools from pseudodifferential calculus.

What carries the argument

The Hodge decomposition for differential k-forms in Sobolev spaces on complete Riemannian manifolds, which isolates the physical transverse components to achieve complete gauge fixing at initial data.

If this is right

  • The Cauchy problem for the gauge-fixed Maxwell equations becomes well-posed on globally hyperbolic spacetimes.
  • Hadamard states exist for the algebra of observables of the quantized Maxwell field after gauge fixing.
  • The algebraic approach to quantum field theory applies directly to the physical degrees of freedom without residual gauge freedom.
  • The same initial-data gauge-fixing technique extends to the general linear gauge theories analyzed earlier in the thesis.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The pseudodifferential calculus employed here aligns naturally with microlocal techniques for tracking wavefront sets in the quantum theory.
  • If the decomposition commutes with the evolution, the resulting states remain Hadamard under time propagation on the full spacetime.
  • The method for handling constrained systems at initial data may inform quantizations of other first-order constrained theories such as linearized gravity.

Load-bearing premise

A Hodge decomposition for k-forms must exist in the required Sobolev spaces on every complete Riemannian manifold that arises as a Cauchy surface of a globally hyperbolic spacetime, and this decomposition must commute appropriately with the hyperbolic evolution to preserve the Hadamard property.

What would settle it

A counterexample consisting of one globally hyperbolic spacetime whose Cauchy surface is a complete Riemannian manifold on which the Sobolev-space Hodge decomposition for 1-forms fails to exist or to be compatible with the evolution operator, or a state constructed by the method that violates the microlocal spectrum condition.

Figures

Figures reproduced from arXiv: 2604.27026 by Gabriel Schmid.

Figure 1.1
Figure 1.1. Figure 1.1: Finite Speed of Propagation and Uniqueness. view at source ↗
Figure 1.2
Figure 1.2. Figure 1.2: Sketch for the definition of Σˆ and Σ ′ . It remains to get rid of the support assumptions on Cauchy data: let (f, ϕ) ∈ Γ∞(Σ0, E|Σ0 )× Γ∞(M, E) be arbitrary and consider an exhaustion of M by compact sets, i.e. a sequence of subsets (Kn)n∈N with Kn ⊂ M compact, such that Kn ⊂ K ◦ n+1 for all n ∈ N. Moreover, choose view at source ↗
Figure 1.3
Figure 1.3. Figure 1.3: A schematic sketch of the support properties of the two classes of nonlocal potentials. view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: Lorentzian metrics g, g ′ and h such that h ⪯ g and h ⪯ g ′ . Having established this result, it is straightforward to show that any globally hyperbolic metric g is paracausally related to an ultrastatic one, see [334, Cor. 2.23]. We identify M = R×Σ and g = −dt ⊗ dt + ht for some time-dependent Riemannian metric ht . This is not a loss of generality, since we can always normalise a globally hyperbolic i… view at source ↗
read the original abstract

This thesis is devoted to the study of hyperbolic differential operators on globally hyperbolic manifolds, linear gauge theories and their quantisation. In the first part, we treat the Cauchy problem for symmetric hyperbolic systems and normally hyperbolic operators on globally hyperbolic manifolds from first principles. Although hyperbolic equations are usually studied with local interactions, there are strong motivations from several areas of mathematical physics to consider also nonlocal interactions. As an intermezzo, we therefore take a small deviation from the classical local theory and prove well-posedness of the Cauchy problem for symmetric hyperbolic systems coupled to a broad class of nonlocal potentials. The second part presents a detailed exposition of linear gauge theories in globally hyperbolic spacetimes. Linear gauge theories are yet another deviation from the concept of hyperbolicity: the corresponding equations of motion are generically non-hyperbolic; however, can always be reduced to a constrained hyperbolic dynamics once an appropriate gauge fixing procedure has been applied. We give a thorough analysis of their Cauchy problem and classical phase space, complemented by a detailed discussion of many examples of physical interest, and discuss their quantisation following the algebraic approach to quantum field theory. The final chapter is devoted to the quantisation of Maxwell's theory on globally hyperbolic spacetimes, with the goal of proving the existence of Hadamard states. The novelty of our approach lies in a new gauge-fixing procedure at the level of initial data, which allows us to suppress the unphysical degrees of freedom. This gauge is achieved by means of a new Hodge decomposition for differential k-forms in Sobolev spaces on complete Riemannian manifolds, while states are constructed using tools from pseudodifferential calculus.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. This thesis studies hyperbolic differential operators on globally hyperbolic manifolds, the classical analysis of linear gauge theories, and their quantisation in the algebraic QFT framework. It first establishes well-posedness for symmetric hyperbolic systems (including nonlocal potentials) and normally hyperbolic operators from first principles. It then reduces linear gauge theories to constrained hyperbolic dynamics via gauge fixing and analyzes their Cauchy problem and phase space. The final chapter constructs Hadamard states for Maxwell theory on globally hyperbolic spacetimes by a new initial-data gauge fixing implemented through a claimed novel Hodge decomposition of k-forms in Sobolev spaces on complete Riemannian manifolds, followed by pseudodifferential calculus to produce the states.

Significance. If the Hodge decomposition exists in the required Sobolev regularity on arbitrary complete Riemannian Cauchy surfaces and commutes with the hyperbolic evolution while preserving the wavefront-set condition, the construction would supply a rigorous, complete gauge fixing that eliminates residual unphysical modes and yields Hadamard states for Maxwell theory. This would strengthen the algebraic quantisation programme for gauge fields on curved backgrounds and could extend to other linear gauge theories.

major comments (2)
  1. [Chapter on Maxwell quantisation / Hodge decomposition statement] The central existence claim for the new Hodge decomposition of k-forms in Sobolev spaces on general complete Riemannian manifolds (without compactness or bounded-geometry assumptions) is load-bearing for the gauge-fixing procedure and the subsequent Hadamard-state construction. The manuscript must supply a self-contained proof that the decomposition is orthogonal, bounded in the precise Sobolev norms used for the initial-value formulation, and correctly projects onto the physical (constraint-satisfying) subspace; standard Hodge theory does not guarantee this on arbitrary complete manifolds.
  2. [Section on evolution of gauge-fixed data and Hadamard property] The commutation of the resulting projection operators with the hyperbolic evolution operator must be verified in detail. Without explicit control on how the projections interact with the evolution, it is unclear whether the gauge-fixed initial data remain in the domain that produces Hadamard two-point functions under the pseudodifferential construction; a wavefront-set argument or explicit commutation relation is required.
minor comments (2)
  1. Notation for the Sobolev spaces and the precise regularity indices used in the initial-data formulation should be stated uniformly across chapters to avoid ambiguity when comparing the classical and quantum parts.
  2. A short table or diagram summarising the reduction steps from the original gauge theory to the final constrained hyperbolic system would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which help clarify the presentation of the novel gauge-fixing procedure. We address each major comment below and will incorporate the suggested clarifications and expansions in the revised version.

read point-by-point responses

  1. Referee: [Chapter on Maxwell quantisation / Hodge decomposition statement] The central existence claim for the new Hodge decomposition of k-forms in Sobolev spaces on general complete Riemannian manifolds (without compactness or bounded-geometry assumptions) is load-bearing for the gauge-fixing procedure and the subsequent Hadamard-state construction. The manuscript must supply a self-contained proof that the decomposition is orthogonal, bounded in the precise Sobolev norms used for the initial-value formulation, and correctly projects onto the physical (constraint-satisfying) subspace; standard Hodge theory does not guarantee this on arbitrary complete manifolds.

    Authors: We agree that the existence and properties of this decomposition are central and that a fully self-contained proof is required. The manuscript already contains a proof of the decomposition for k-forms in Sobolev spaces on complete Riemannian manifolds, relying on the elliptic theory of the Hodge Laplacian and the completeness assumption to establish the necessary estimates. To address the referee's request explicitly, we will expand the relevant section (in the chapter on Maxwell quantisation) with a more detailed, self-contained argument verifying orthogonality in the L2 inner product, boundedness in the precise Sobolev norms appearing in the initial-value formulation, and the fact that the projection lands in the constraint-satisfying subspace. We will also clarify why the completeness assumption suffices without invoking bounded geometry. revision: yes

  2. Referee: [Section on evolution of gauge-fixed data and Hadamard property] The commutation of the resulting projection operators with the hyperbolic evolution operator must be verified in detail. Without explicit control on how the projections interact with the evolution, it is unclear whether the gauge-fixed initial data remain in the domain that produces Hadamard two-point functions under the pseudodifferential construction; a wavefront-set argument or explicit commutation relation is required.

    Authors: We thank the referee for this observation. The manuscript shows that the gauge-fixed initial data evolve while preserving the constraints, which is used to ensure the Hadamard property via the pseudodifferential construction. However, we acknowledge that an explicit verification of commutation would strengthen the argument. In the revised version we will add a dedicated subsection deriving the commutation relation between the Hodge projections and the hyperbolic evolution operator, together with a wavefront-set analysis confirming that the resulting two-point functions remain in the Hadamard class. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via independent proofs

full rationale

The paper constructs a gauge-fixing procedure for Maxwell theory using a claimed new Hodge decomposition of k-forms in Sobolev spaces on complete Riemannian manifolds, followed by pseudodifferential calculus to obtain Hadamard states. This relies on standard functional-analytic existence results and the algebraic QFT framework rather than any self-referential definitions, fitted parameters renamed as predictions, or load-bearing self-citations. The central steps (decomposition existence, orthogonality, and evolution commutation) are presented as proven from first principles, not reduced to the inputs by construction. No patterns from the enumerated circularity kinds appear.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The work rests on standard background results in differential geometry and functional analysis rather than introducing new free parameters or postulated entities.

axioms (3)
  • domain assumption Globally hyperbolic Lorentzian manifolds admit Cauchy surfaces that are complete Riemannian manifolds.
    Invoked throughout the treatment of the Cauchy problem and the Hodge decomposition.
  • domain assumption Symmetric hyperbolic systems with nonlocal potentials admit unique solutions to the Cauchy problem on globally hyperbolic manifolds.
    Used in the intermezzo on nonlocal interactions.
  • domain assumption The algebraic approach to QFT supplies a consistent quantization once a suitable symplectic space of solutions is identified.
    Basis for the final quantization step.

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Reference graph

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